Extended Abstract

1. Distribution of Groups of Vertices
Across Multilayer Networks
With an Application to 3–Layer Twitter Networks
Extended Abstract
Moses A. Boudourides1
& Sergios T. Lenis2
Department of Mathematics
University of Patras, Greece
1Moses.Boudourides@gmail.com
2sergioslenis@gmail.com
1 March 2015
M.A. Boudourides & S.T. Lenis Distribution of Groups of Vertices Across Multilayer Networks

2. Distribution of Groups of Vertices
Across Multilayer Networks
With an Application to 3–Layer Twitter Networks
Extended Abstract
Moses A. Boudourides1
& Sergios T. Lenis2
Department of Mathematics
University of Patras, Greece
1Moses.Boudourides@gmail.com
2sergioslenis@gmail.com
1 March 2015
M.A. Boudourides & S.T. Lenis Distribution of Groups of Vertices Across Multilayer Networks

3. Abstract
Here, we understand a multilayer network composed of m(≥ 2) layers as
an arrangement of m graphs or digraphs, which are joined together
through an m–partite graph. One may consider an equivalent
formulation, motivated by balance theory of signed graphs, according to
which a multilayer network is a “balanced” edge–colored graph with
regards to a cut such that edges in each block of the cut take one of m
different colors and all the cut–edges have a different (m + 1)–th color.
Our aim here is to study how edge colors are distributed over certain
structural groups of graph vertices, such as connected components and
(modularity maximizing) communities. In this way, given a subgraph in a
multilayer network, the subgraph is said homogeneous (or monolayered) if
all its vertices belong to the same layer and, otherwise, the subgraph is
called mixed (or polylayered). As an application of the study of layered
groups of vertices, we are analyzing a 3–layer network extracted from
Twitter data (about 500 K tweets in the period October 18–31, 2013,
retrieved through the search term“Obamacare”) with the following layers:
(i) the layer of retweets among Twitter users, (ii) the layer of following
relationships among these users and (iii) the layer of co-occurring
hashtags included in the tweets sent by the Twitter users.
M.A. Boudourides & S.T. Lenis Distribution of Groups of Vertices Across Multilayer Networks

4. Basic Definitions and Concepts
Here, we understand a multilayer network composed of m(≥ 2) layers as an
arrangement of m graphs or digraphs, which are joined together through an
m–partite graph.
Sometimes (cf. Roberts, 1999), a signed graph is considered as an edge–colored
graph with just two colors that are expressed in terms of two signs (the + and
the − sign).
Moreover, a signed graph is balanced (Harary, 1954) whenever one of the two
edge colors (the − sign) is assigned to the cutset corresponding to a partition
(cut) (V1, V2) of the graph vertices such that all edges in the subgraphs induced
by V1 and V2 possess the second color (the + sign).
Definition
A graph G = (V , E) is said to be a balanced edge–colored graph,
whenever, for m ≥ 2, there exists a cut (V1, V2, . . . , Vm) of V such
that every edge in E is colored in one of the m + 1 colors
Cm+1 = (c1, c2, . . . , cm, cm+1) in the following way:
(i) for any k = 1, 2, . . . , m, all edges in the subgraph induced by
Vk take the color ck and
(ii) all cut–edges in the cut (V1, V2, . . . , Vm) take the color cm+1.
M.A. Boudourides & S.T. Lenis Distribution of Groups of Vertices Across Multilayer Networks

5. Definition
A multilayer network is a balanced edge–colored graph with regards to a
cut such that edges in each block of the cut take one of m different
colors and all the cut–edges have a different (m + 1)–th color.
Let G be an edge–colored graph, each edge of which takes a color from a set of
colors Cm+1. Then, given a subgraph F of G and a color ck ∈ Cm+1, we denote
by εk (F) the number of edges in F, which are colored in ck .
We will say that two colors cp, cq ∈ Cm+1 are adjacent colors if there exists a
vertex v and two (distinct) edges incident to v such that one of these edges is
color in cp and the other is colored in cq.
Proposition
Let G be an edge–colored graph, each edge of which is colored in one of
the m + 1 colors of the set Cm+1, where m ≥ 2. Then G is a balanced
edge–colored graph if and only if, there exists a color c0 ∈ Cm+1 such
that, for any cycle C in G,
(i) ε0(C) is either an even nonnegative integer, when m = 2, or it is
any nonnegative integer different than 1, when m ≥ 3, and,
(ii) for any cp, cq ∈ Cm+1 {c0}, cp = cq, the colors cp, cq are not
adjacent colors in C.
M.A. Boudourides & S.T. Lenis Distribution of Groups of Vertices Across Multilayer Networks

6. Definition
Let K be a subgraph of a multilayer network. Then K is said to
be:
homogeneous (or monolayered) if all vertices in K belong to
the same layer, and,
otherwise, K is called mixed (or polylayered).
Definition
Let G = (V , E) be a multilayer network and C = {C1, C2, . . . , Cc}
be a partition of V into c subsets (groups) of vertices. Then C is
said to be:
homogeneous (or monolayered) if, for any j = 1, . . . , c, the
subgraph G(Cj ) (induced by Cj ) is homogeneous, and,
otherwise, C is called mixed (or polylayered).
M.A. Boudourides & S.T. Lenis Distribution of Groups of Vertices Across Multilayer Networks